A note on the Lovász theta number of random graphs
Sanjeev Arora
Aditya Bhaskara
Abstract
We show a strong concentration bound for the Lovász ϑ function on G(n, p) random graphs.
For p = 1/2, for instance, our result implies that the ϑ function is concentrated in an interval
of length polylog(n) w.h.p. The best known bound previously was roughly n 1/4 . The general
idea is to prove that all the vectors in an optimal solution have “roughly equal lengths” w.h.p.
1
Introduction
The Lovász ϑ function of a graph is a quantity introduced by Lovász to study the Shannon capacity
of a graph [4]. It is a semidefinite programming relaxation for the independent set of a graph. For
a graph G = (V, E), it is formally defined as follows (see [4] for other equivalent formulations)
X
vi · v0 s.t.
ϑ(G) := max
i
vi2
v02
= v i · v0
=1
∀i
hvi , vj i = 0 ∀{i, j} ∈ E(G)
The expected value of the Lovász ϑ function for G(n, p) random graphs was first studied by
Juhász [3], who showed that for G ∼ G(n, p) and p ≥ log 2 n/n, we have
r
r
n
n
1
≤ ϑ(G) ≤ 2
w.p. at least 1 − .
p
p
n
More recently, Coja-Oghlan studied the concentration properties of the ϑ function for G(n, p)
random graphs [2]. He proved that the ϑ function is concentrated in intervals of length O(1) w.h.p.
when p < n−1/2 . More precisely, he proves the following large deviation bound for ϑ(G): suppose
G ∼ G(n, p) and let µ be the median value of ϑ(G). Then
Pr[|ϑ(G) − µ| > t] ≤ e−t
2 /(µ+t)
.
Note that for say p = 1/2, this only says that ϑ(G) is concentrated in an interval of length roughly
n1/4 w.h.p.1 In this note, we will show a better tail bound. More precisely,
Theorem 1. Let G = (V, E) be a graph drawn from G(n, 1/2). Let µ denote the median of ϑ(G)
for this distribution. Then for some absolute constant C, we have
Pr[|ϑ(G) − µ| > t] ≤ e−t
1
4/3 /(C
Throughout, when we say “w.h.p.”, we mean w.p. at least 1 −
parameters which naturally depend on c).
1
log3 n)
1
nc
,
(1)
for any constant c (there will be certain
Our techniques are not specific to p = 1/2, but for ease of exposition, we will only work with
this case. This implies, for instance, that for G(n, 1/2) random graphs, ϑ(G) is concentrated in
intervals of size only polylog(n).
Comment. The exponent 4/3 is unnatural, and we believe it is an artefact of our proof – we
2
conjecture that the “true” tail bound is in fact (1) with e−t /C log n on the RHS.
2
Proof
In what follows, let µ denote the median of ϑ(G) for G ∼ G(n, 1/2), and let t be a given parameter.
Let s be a parameter (we will set it to be max{t2/3 , log n}). A graph G is said to be s-bad if for all
vector solutions vi which “realize” the optimum value for the relaxation ϑ(G), we have
X
kvi k4 > (1 + s) log2 n.
i∈V
Lemma 2. Suppose G is s-bad for some s ≥ log
pn. Then there exists an S ⊆ V of size k ≥ s such
that the induced subgraph H on S has ϑ(H) > k(1 + s) log n.
Proof. Let {vi }ni=1 denote an optimum vector solution for the ϑ relaxation on G. It is easy to see
1
(we
that there exists a solution with value at least ϑ(G)/2 and the additional property kv i k2 ≥ 2n
can simply set vectors which are smaller than this length to zero). Now divide the v i into log n
levels based on kvi k2 , such that thePvalue of kvi k2 varies by a factor at most 2 in each level.
Since G is s-bad, we have that i kvi k4 ≥ (1 + s) log2 n. There exists a level which contributes
at least a 1/ log n fraction to the sum: let S be the set of indices in this level, and let k = |S|. Thus
1/2
p
P
for each i ∈ S, we have kvi k2 ≈ (1+s)k log n
, implying that i∈S vi2 ≥ 12 k(1 + s) log n. Since
vi is a feasible solution to the relaxation
ϑ(G), it is clear that the restriction to S gives a feasible
p
solution to ϑ(H). Thus ϑ(H) ≥ k(1 + s) log n.
Finally, since kvi k2 ≤ 1, we must have k ≥ s, thus proving the lemma.
We can now bound the probability that G ∼ G(n, 1/2) is s-bad for some s ≥ log n. Fix some
set S ⊆ V of size k and let H be the induced subgraph on S in G. We now use a bound of [4]
relating ϑ(H) to the eigenvalues of its adjacency matrix.
Lemma 3. [4] Let G be a graph with adjacency matrix A(G), J denote the n × n matrix of ones,
and I the identity matrix. Then
ϑ(G) ≤ λmax (J − 2A(G) − I).
We refer to the paper of Lovász for the proof [4]. It follows from one of the equivalent definitions
of the ϑ function. The second ingredient is a concentration bound for the top eigenvalues of a
random matrix due to Alon, Krivelevich and Vu [1]. They prove the following.
Lemma 4. Let A be a symmetric n × n matrix with the upper diagonal entries drawn i.i.d. from
a distribution with mean zero and variance 1. Then for all t > 0, and integer r ≥ 1, we have
Pr[|λr (A) − µ(λr (A))| ≥ t] ≤ e−t
2 /2r 2
.
(2)
(As usual λr denotes the rth largest eigenvalue, and µ(λr ) denotes the median of this value over
the distribution)
2
Now we note that for any fixed S ⊆ V of size k, the matrix J − 2A(H) − I is a k × k symmetric
matrix with entries being i.i.d. √±1 (and zero on the diagonal). Thus the median of λ max (J −
2A(G) − I) is at most (2 + o(1)) k, and by Lemma 4, we have
p
Pr λmax (A(H)) > k(1 + s) log n < e−k(1+s) log n .
p
k(1 + s) log n is also bounded by the same
Now by Lemma 3, the probability that ϑ(H) >
quantity. Thus we can take a union bound over all subsets of size k ≥ s, and by Lemma 2, we have
X
X n
2
e−sk log n ≤ e−s log n .
· e−(1+s)k log n <
Pr[G is s-bad] ≤
k
k≥s
k≥s
n
(In the above we used k ≥ s, and a simple bound on k ). We have thus proved that
Lemma 5. Let G ∼ G(n, 1/2), and s ≥ log n. The probability that G is s-bad is at most e −s
2
log n .
We can now follow the proof of Coja-Oghlan [2] (and [1]) and use Talagrand’s inequality. Let
us first recall it.
Theorem 6. (Talagrand)[5] Let Ω be a set with a measure µ defined on it, and let A, B ⊆ Ω n .
Let µn denote the product measure obtained from µ. Suppose A
Pand B are “t-separated” in the
following way: for every b ∈ B, there exist weights {αi }ni=1 with i αi2 ≤ 1 such that
X
αi ≥ t.
∀ a ∈ A,
i:ai 6=bi
Then we have µn (A)µn (B) ≤ e
−t2
.
The theorem is very powerful, and we typically use it with finite sets Ω. Let us now define two
sets of graphs as follows
A := {G : ϑ(G) ≤ µ}, and
B := {G : ϑ(G) ≥ µ + t, and G is not s-bad for s = max{t2/3 , log n}}.
Let m(A) (similarly B) denote the measure of A in the set of graphs G(n, 1/2). Since µ was
defined to be the median, m(A) = 1/2.
Lemma 7.
m(A) · m(B) ≤ e−t
2 /(1+s) log n
.
Proof. Consider a graph B ∈ B. Let {vi }ni=1 be the set of vectors in an optimal solution to the
ϑ-relaxation on B. Now consider any A ∈ A.
Let αi be 1 if vertex i has precisely the same set of neighbors in both A and B, and 0 otherwise. Now observe that {αi vi } is a feasible vector solution to the ϑ relaxation
for B (because
P
2
αi αj 6= 0 implies {i, j} is an edge in B iff it is an edge in A). Thus
i (αi vi ) ≤ ϑ(B), hence
P
2 ≥ t (since ϑ(B) < µ).
v
i:ΓA (i)6=ΓB (i) i
P 2 2
2
Now by the definition of B, B is not s-bad, hence we have
i (vi ) ≤ (1 + s) log n. By
2
Talagrand’s inequality, we have
m(A) · m(B) ≤ e−t
2 /(1+s) log2
n
.
Formally, the product space here is Ωn , where Ω consists of vectors in {0, 1}n representing the adjacency vectors
of a vertex in the graph. In these terms, αi is an indicator for the ith vectors corresponding to A, B being equal.
2
3
Corollary 8. Let G ∼ G(n, 1/2). Then
Pr[ϑ(G) > µ + t] ≤ e−t
4/3 / log2
n
.
Proof. From the above lemmas, we can bound the desired probability by
Pr[G is s-bad] + Pr[G ∈ B]
2
= e−s + e−t
2 /(1+s) log2
n
≤ e−t
4/3 / log3
n
.
The last inequality is due to our choice of s.
Lower tail.
A bound for the lower tail is actually easier to prove: as before, define two sets
A := {G : ϑ(G) ≤ µ − t}, and
B := {G : ϑ(G) ≥ µ, and G is not log n-bad}.
2
The key is to note that the probability that G is log n-bad is only e− log n 1/10, and thus
m(B) ≥ 1/3 (because without this restriction, the measure is 1/2, since µ is the median). Now
using precisely the same argument as above, we obtain
m(A) ≤ e−t
2 / log3
n
.
This completes the proof of Theorem 1.
References
[1] Noga Alon, Michael Krivelevich, and Van Vu, On the concentration of eigenvalues of random
symmetric matrices, Israel Journal of Mathematics 131 (2002), 259–267, 10.1007/BF02785860.
[2] Amin Coja-Oghlan, The lovsz number of random graphs, Approximation, Randomization, and
Combinatorial Optimization.. Algorithms and Techniques (Sanjeev Arora, Klaus Jansen, Jos
Rolim, and Amit Sahai, eds.), Lecture Notes in Computer Science, vol. 2764, Springer Berlin /
Heidelberg, 2003, pp. 5–19.
[3] Ferenc Juhász, The asymptotic behaviour of lovász’ theta-function for random graphs, Combinatorica 2 (1982), no. 2, 153–155.
[4] L. Lovasz, On the shannon capacity of a graph, Information Theory, IEEE Transactions on 25
(1979), no. 1, 1 – 7.
[5] Michel Talagrand, Concentration of measure and isoperimetric inequalities in product spaces,
Publications Mathmatiques de L’IHS 81 (1995), 73–205, 10.1007/BF02699376.
4